### Abstract

We consider the expressive power of the first-order structure 〈Ω,C 〉 where Ω is either of two of different domains of extended regions in Euclidean space, and C(x,y) is the topological relation 'Region x is in contact with region y.' We prove two main theorems: Let P[+] be the domain of bounded, non-empty, rational polyhedra in two- or three-dimensional Euclidean space. A relation Γ over P[+] is definable in the structure 〈P[+], C〉 if and only if Γ is arithmetic and invariant under rational PL-homeomorphisms of the space to itself. We also extend this result to a number of other domains, including the domain of all polyhedra and the domain of semi-algebraic regions.Let R be the space of bounded, non-empty, closed regular regions in n-dimensional Euclidean space. Any analytical relation over lower dimensional (i.e. empty interior) compact point sets that is invariant under homeomorphism is implicitly definable in the structure 〈R,C〉.

Original language | English (US) |
---|---|

Pages (from-to) | 1107-1141 |

Number of pages | 35 |

Journal | Journal of Logic and Computation |

Volume | 23 |

Issue number | 5 |

DOIs | |

State | Published - Oct 2013 |

### Fingerprint

### Keywords

- expressivity
- First-order language
- qualitative spatial reasoning
- topological language

### ASJC Scopus subject areas

- Logic
- Theoretical Computer Science
- Software
- Hardware and Architecture
- Arts and Humanities (miscellaneous)

### Cite this

**The expressive power of first-order topological languages.** / Davis, Ernest.

Research output: Contribution to journal › Article

*Journal of Logic and Computation*, vol. 23, no. 5, pp. 1107-1141. https://doi.org/10.1093/logcom/ext002

}

TY - JOUR

T1 - The expressive power of first-order topological languages

AU - Davis, Ernest

PY - 2013/10

Y1 - 2013/10

N2 - We consider the expressive power of the first-order structure 〈Ω,C 〉 where Ω is either of two of different domains of extended regions in Euclidean space, and C(x,y) is the topological relation 'Region x is in contact with region y.' We prove two main theorems: Let P[+] be the domain of bounded, non-empty, rational polyhedra in two- or three-dimensional Euclidean space. A relation Γ over P[+] is definable in the structure 〈P[+], C〉 if and only if Γ is arithmetic and invariant under rational PL-homeomorphisms of the space to itself. We also extend this result to a number of other domains, including the domain of all polyhedra and the domain of semi-algebraic regions.Let R be the space of bounded, non-empty, closed regular regions in n-dimensional Euclidean space. Any analytical relation over lower dimensional (i.e. empty interior) compact point sets that is invariant under homeomorphism is implicitly definable in the structure 〈R,C〉.

AB - We consider the expressive power of the first-order structure 〈Ω,C 〉 where Ω is either of two of different domains of extended regions in Euclidean space, and C(x,y) is the topological relation 'Region x is in contact with region y.' We prove two main theorems: Let P[+] be the domain of bounded, non-empty, rational polyhedra in two- or three-dimensional Euclidean space. A relation Γ over P[+] is definable in the structure 〈P[+], C〉 if and only if Γ is arithmetic and invariant under rational PL-homeomorphisms of the space to itself. We also extend this result to a number of other domains, including the domain of all polyhedra and the domain of semi-algebraic regions.Let R be the space of bounded, non-empty, closed regular regions in n-dimensional Euclidean space. Any analytical relation over lower dimensional (i.e. empty interior) compact point sets that is invariant under homeomorphism is implicitly definable in the structure 〈R,C〉.

KW - expressivity

KW - First-order language

KW - qualitative spatial reasoning

KW - topological language

UR - http://www.scopus.com/inward/record.url?scp=84885123257&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84885123257&partnerID=8YFLogxK

U2 - 10.1093/logcom/ext002

DO - 10.1093/logcom/ext002

M3 - Article

VL - 23

SP - 1107

EP - 1141

JO - Journal of Logic and Computation

JF - Journal of Logic and Computation

SN - 0955-792X

IS - 5

ER -